I think I've really screwed up

Talk to her and be very clear about your points you posted here. Stay confident. Talk to your aunt and your mother about it. Do not talk about your schedule during recruitment to her or anyone else. Just stay open minded, have a smile, write down your own decisions in another room and if it gets worse try to get another roommate. Do not speak about it during rounds it might look desperate, talk to your RA and rho chi.

Stay positive. Maybe your cousin is just really excited because she is going to experience this all with you. I think you should just move in and start the process with a smile and an open mind. Keep as much as you can to yourself. Both of you will end up in a house that I hope you each love. Through this process you both will find your own circle of friends. There is so much that can happen in the first semester and she just might want to room with a sorority sister and not you going forward. Don't stress now and don't let this ruin your moment. You only get this experience one time in your life.

NEVER EVER let anyone get the idea that you are a package deal. It will ruin rush for both of you and cut your options way down. Talk to your cousin and tell her that both of you need to keep your mouths shut for both your sakes and that she needs to rush FOR HERSELF alone and not even mention you, her cousin and the same for you. If she doesn't seem to get it then your Rho Chi needs to step in and she can hear it from her. Take what @Tricky stuff says to heart and do not share your sorority thoughts and opinions with anyone. The odds of you both getting the same house are not high anyway, and it happens to twin sisters and very close BFF's. But actually it's a blessing because you each can remain close and do stuff together anyway, and while experiencing your own houses you will be welcome at events at the other's house and make even more friends. Win-Win! Rush is an emotional time and double so when close friends become roommates; this is not going to be like rooming together at summer camp.